# The proof of the Nirenberg-Treves conjecture

• page 1-25
• ISSN: 0752-0360

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## Abstract

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We prove the Nirenberg-Treves conjecture : that for principal type pseudo-differential operators local solvability is equivalent to condition ($\Psi$). This condition rules out certain sign changes of the imaginary part of the principal symbol along the bicharacteristics of the real part. We obtain local solvability by proving a localizable estimate for the adjoint operator with a loss of two derivatives (compared with the elliptic case). The proof involves a new metric in the Weyl (or Beals-Fefferman) calculus. This makes it possible to reduce to the case when the gradient of the imaginary part is non-vanishing, and then the zeroes form a smooth submanifold. The estimate uses a new type of weight, which measures the change of the distance to the zeroes of the imaginary part along the bicharacteristics of the real part between the minima of the curvature of this submanifold. By using condition ($\Psi$) and this weight, we can construct a multiplier which gives the estimate.

## How to cite

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Dencker, Nils. "The proof of the Nirenberg-Treves conjecture." Journées équations aux dérivées partielles (2003): 1-25. <http://eudml.org/doc/93447>.

@article{Dencker2003,
abstract = {We prove the Nirenberg-Treves conjecture : that for principal type pseudo-differential operators local solvability is equivalent to condition ($\Psi$). This condition rules out certain sign changes of the imaginary part of the principal symbol along the bicharacteristics of the real part. We obtain local solvability by proving a localizable estimate for the adjoint operator with a loss of two derivatives (compared with the elliptic case). The proof involves a new metric in the Weyl (or Beals-Fefferman) calculus. This makes it possible to reduce to the case when the gradient of the imaginary part is non-vanishing, and then the zeroes form a smooth submanifold. The estimate uses a new type of weight, which measures the change of the distance to the zeroes of the imaginary part along the bicharacteristics of the real part between the minima of the curvature of this submanifold. By using condition ($\Psi$) and this weight, we can construct a multiplier which gives the estimate.},
author = {Dencker, Nils},
journal = {Journées équations aux dérivées partielles},
keywords = {local solvability; Nirenberg-Treves conjecture; Weyl calculus; pseudo-differential operator},
language = {eng},
pages = {1-25},
publisher = {Université de Nantes},
title = {The proof of the Nirenberg-Treves conjecture},
url = {http://eudml.org/doc/93447},
year = {2003},
}

TY - JOUR
AU - Dencker, Nils
TI - The proof of the Nirenberg-Treves conjecture
JO - Journées équations aux dérivées partielles
PY - 2003
PB - Université de Nantes
SP - 1
EP - 25
AB - We prove the Nirenberg-Treves conjecture : that for principal type pseudo-differential operators local solvability is equivalent to condition ($\Psi$). This condition rules out certain sign changes of the imaginary part of the principal symbol along the bicharacteristics of the real part. We obtain local solvability by proving a localizable estimate for the adjoint operator with a loss of two derivatives (compared with the elliptic case). The proof involves a new metric in the Weyl (or Beals-Fefferman) calculus. This makes it possible to reduce to the case when the gradient of the imaginary part is non-vanishing, and then the zeroes form a smooth submanifold. The estimate uses a new type of weight, which measures the change of the distance to the zeroes of the imaginary part along the bicharacteristics of the real part between the minima of the curvature of this submanifold. By using condition ($\Psi$) and this weight, we can construct a multiplier which gives the estimate.
LA - eng
KW - local solvability; Nirenberg-Treves conjecture; Weyl calculus; pseudo-differential operator
UR - http://eudml.org/doc/93447
ER -

## References

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12. [12] Lars Hörmander, On the solvability of pseudodifferential equations, Structure of solutions of differential equations (M. Morimoto and T. Kawai, eds.), World Scientific, New Jersey, 1996, pp. 183-213. Zbl0897.35082MR1445340
13. [13] Nicolas Lerner, Sufficiency of condition (Ψ) for local solvability in two dimensions, Ann. of Math. 128 (1988), 243-258. Zbl0682.35112MR960946
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16. [16] Nicolas Lerner, Perturbation and energy estimates, Ann. Sci. École Norm. Sup. 31 (1998), 843-886. Zbl0927.35139MR1664214
17. [17] Nicolas Lerner, Factorization and solvability, Preprint.
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20. [20] Jean-Marie Trépreau, Sur la résolubilité analytique microlocale des opérateurs pseudodifférentiels de type principal, Ph.D. thesis, Université de Reims, 1984.

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